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October 5, 2025Open Access

Infinitely presented simple groups separated by homological finiteness properties

Key Points

A simple group is constructed that resembles finitely generated linear groups, and serves as a quasi-retract.
It demonstrates the existence of simple groups of type FP_infinity that are not finitely presented.
For every natural number n, a simple group of type FP_n is shown to exist that does not meet the criteria for FP_n+1.
These findings provide a solution to a previously posed question regarding Röver–Nekrashevych groups.

Abstract

Given a finitely generated linear group G over Q, we construct a simple group Γ that has the same finiteness properties as G and admits G as a quasi-retract. As an application, we construct a simple group of type FP_ that is not finitely presented. Moreover we show that for every n N there is a simple group of type FPₙ that is neither finitely presented nor of type FP₍+₁. Since our simple groups arise as Röver--Nekrashevych groups, this answers a question of Zaremsky.

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